Explanation

15 = 15 or 5×3, so N = p^14 or p⁴q². But p^14 ≥ 2^14 = 16384 is 5-digit, impossible for 4-digit N; hence N = p⁴q². Then N² = p⁸q⁴ has (8+1)(4+1) = 45 divisors — option C.